A Problem Is a Problem for All That

Stage: 2, 3 and 4

This article was first published
in The Scottish Mathematical Council's Journal (2008), Vol 38,
P54-57.

Often, when I start a problem-solving session I begin by saying
something like:

"Today we are going to be problem solving and this means working on
problems. So, before we start it might be good to know what I mean
by a problem. A problem is something that you probably will not
immediately see how to tackle or which once started will challenge
your thinking. This means that at some point you are likely to get
stuck and this is OK. In fact it is important that you get stuck so
that we can talk about possible ways of getting unstuck."

In other words the very problem with problems, namely that they
should result in you being stuck, is at the heart of what problem
solving is about. This contrasts strongly with most classroom
experiences of learners when being stuck is equated with failure.
In this article I am going to talk about just a few of the other
problems with problems that make them such a rich source of
mathematics and approaches to learning mathematics.

Of course, I have immediately set myself a problem because whatever
'problems' I select for us to work on here, they will not
necessarily be problems for all of you. They may not challenge you
all (especially if you have seen them, or questions like them,
before and therefore you know what to do and do not get stuck). I
will do my best to identify good problems that offer interesting
insights and the rest is up to you. So, in discussing what the
problem with problems might be, I will try to select examples that
will engage you and extend your thinking and perhaps even surprise
you. Even if you have seen a problem before and think you 'know
it', I challenge you to ask yourself "What next?" or "What if?" or
"Do I really know all there is to be known about this problem?" or
"Can I learn more?".

So this article is as much about doing as reading. In the true
spirit of problem solving though, I am trying to seed some ideas
and leave you to try things out and refine your own thinking. I
cannot be sure where you will end up but I have some sense of the
story I am trying to tell, and that is about some of the problems
that arise from working with problems and how these are all part of
being mathematical.

Time and Tide

Have you seen something like this before?
Do you immediately know what to do?
Are you fairly confident you can quickly find the answer?

If your answer to two or more of the above is "Yes", as it stands,
this is probably not a problem for you and you should ask some of
the questions that extend your thinking (for example: What if it
wasn't 3 and 4? Can you generalise? Can you prove ...?).

If you are not certain what to do, spend some time thinking about
this problem. Make yourself a cup of tea, relax, take your time, as
you can always come back to the article later.

If you had no idea where to start, what strategies did you adopt or
did you just give up and decide to read on?

A colleague of mine walked past this problem for nearly a year. It
took him this long before the penny dropped and it 'suddenly' went
from being a problem to not being a problem. So what, if anything,
happened over that year? I would say lots. My colleague allowed
himself all the time he wanted to pull all those ideas and issues
hiding in metaphorical cupboards together. There was no pressure.
No one was testing him. There were no time constraints.

I hope you are thinking how similar to the classroom this
isn't.

So, one problem with problems in the classroom is time. I do not
mean the time to fit problem solving into the curriculum but the
time to go about the activity of problem solving. What strategies
can we bring to bear to deal with this? Well the first thing I am
tempted to say is 'give your pupils time' and allowing 'having
time' to be part of their mathematical experience seems important.
Another colleague on the NRICH team talks about 'pot boilers'. If
one lesson is not long enough to get focussed and thinking, how
about introducing a problem or idea in one lesson and keep on
popping back to see how it is doing? (Has anyone had any ideas
since we last looked at this?). We should welcome things that are
left stated but untried, started but not finished, finished but in
arriving at a solution raise more questions. Why not have a 'pot
boiler' always on your classroom wall and come back to it every so
often to see if anyone has had any ideas. This is basically what my
colleague did with Big Powers.

Another strategy related to time might be to work on things more
collaboratively but with the pupils working on the problem together
with your support. That is, they come up with ideas that you help
them share, stopping occasionally when a new discovery or step has
been made to highlight the progress for all. One strategy that
might be useful here is to encourage learners to share their ideas
with the class, facing the class, rather than addressing them to
you as the teacher, and asking the group to feedback, comment or
ask further questions.

What I am not advocating is breaking the problem down and taking
the learners step by step through the journey. Perhaps if they
cannot suggest, with your support, possible approaches and
strategies and you want them to move on quickly it was not a 'good
problem' and best left for the moment. What will they learn by
simply being told what to do? If they do learn something it is
difficult to see what it has to do with problem solving so what was
the point? Why not make it a pot boiler?

Implicit in what I have been saying, but not yet stated, is that
problems take longer. Learners will inevitably spend longer on each
question that is set. In doing so, learners are given time to
engage in and own the mathematics rather than learning a method
that is easily forgotten and rarely applied.

So one issue is definitely time, but this is a good problem to have
to work with. Many teachers I meet say that they wished their
pupils would not give up so easily, would persevere and spend
longer trying harder and not demanding to know what to do or what
the answer is. What better way to encourage a change in such
attitudes?

Same but Different

Now spend some time thinking about the following problem, called
Sums of Pairs before reading on:

I remember a session when one of my colleagues was working with a
group of Year 10 pupils. In the session the pupils worked in small
groups trying to solve the problem. Probably, like me, your first
approach might be by using algebra but what was interesting was the
different approaches the pupils adopted in order to find a
solution. For example:

some learners used an algebraic approach (calling the original
three numbers a, b and c)

others used trial and improvement. One of these groups starting
by saying that the smallest number has to be $8$ or less in order
to make a total of $17$ (why?) ...

another group of pupils found a solution almost 'by accident',
a set of random attempts result in unexpected success. At that
point my colleague gave them a similar problem to encourage them to
search for a method that they could generalise.

After some time the pupils were encouraged to share their
methods and then to discuss approaches they found most accessible,
repeatable and efficient. Great value was being placed on
creativity, as differences in approaches were described, valued and
assessed. This formed a key part of the learning process. It was
the means by which the pupils made their journeys towards a
solution rather than the solution itself which was being
valued.

Everyone was encouraged to listen to, and learn from others.
Of course, pupils needed to be confident enough to communicate with
and to their peers and come to their own views. This takes time and
practice but is achievable if pupils are used to working in this
way and given lots of opportunity to practise.

In this case the problem with the problem is the lack of
'control' of the way it was tackled. But this is its main strength.
In fact I don't think the algebraic approach is the most elegant in
this case and one of the
solutions on the website is as follows:

I think that is so neat and not at all the way I thought of it. I
certainly learnt a lot and what a useful idea to store (partial
sums). Now you may have done it this way, but watching the
approaches adopted by the pupils may have offered you something
different, not least the way they engaged in each other's
methods.

Perhaps I should end with just one more thought, leaving room for
plenty of other problems to be investigated later!

Outcome Versus Journey

I have mentioned earlier the importance of the journey and
sometimes there is a balance to be struck between letting go and
wanting a particular outcome. A problem that comes to mind in this
respect is
Making Rectangles, Making Squares :

I use this problem a great deal but I never start from the question
given, and only once or twice have the group I have been working
with answered the question as set. This is because so much other
rich mathematics arises naturally from discussion that I find it a
shame to let it go. Also, by inviting learners' own ideas the
problem offers me an opportunity to assess their knowledge as well
as their ability to pose their own problems. At one level, it is
possible to discuss proof, use rational and irrational numbers, as
well as utilise Pythagoras' theorem. At other levels you might find
yourself exploring symmetry or triangle animals (all possible
shapes made from $2$, $3$, $4$ ... triangles that are joined by
common sides). It links to a number of other problems on the NRICH
website site including
Triangle Relations and
Equal Equilateral Triangles .

The tension for me is always about balancing the original
intentions of the problem with what the learners are telling me in
terms of their own understandings. In general I tend to use the
context, rather than the problem itself, as a starting point for
encouraging learners to see the mathematics in a situation and to
pose their own problems.

So how might this happen? Perhaps if I relate how I have used this
problem, things will be clearer. Firstly, I just hand out the
triangles without any indication of how they are related or formed.
Working in small groups the learners 'play'
with the triangles for a few minutes and write on large sheets of
paper (to share with the rest of the group) what they consider to
be four key mathematical properties of the triangles.

The class is then invited to walk around the room and look at what
other groups have written. At this point they can add anything else
they feel is important to their list. Discussion as a whole group
reveals what they were most 'surprised' by or intrigued by as they
walked around and what they might like to investigate further. It
is normally at this point we talk about the relationships between
the two triangles- including things like their equal areas.

By now many of the learners are coming up with problems of their
own, although not yet fully articulated:

"The two triangles can be put together to form a right-angled
triangle. Can we make right-angled triangles any other way?"
"Can we make a square?"
"Can we make a rectangle from just one colour?"
"What sized equilateral triangles can we make?"
"Can we make the same sized equilateral triangle in just red or
just green triangles?"
"What symmetrical shapes can we make?"

The list is almost endless but it is at this point I have to make a
decision; whether to encourage the exploration of different areas
of interest or bring the focus of the lesson in on a particular
idea. I could for example, pick up on the first three questions
posed and suggest an exploration of the problem posed on the site.
There is a great range of possibilities around equilateral
triangles and hexagons and, of course, there are all the problems
based on symmetry (for example, how many different symmetrical
shapes can you make with just four triangles?). I remain very
flexible and tend only to go in a particular direction, such as
squares and rectangles, if it has appeared naturally out of earlier
group discussions from several of the groups. My rationale for this
is that if the groups have not seen the connection naturally they
may not be ready (though I do make efforts to extend thinking as
the groups work).

I know that we often use a problem to bring out a certain aspect of
mathematics, and when this is the case a certain amount of
classroom choreography is needed to provide a level of steer in the
intended direction. However, with this problem I am rarely using it
for the sole purpose of discussing rational and irrational numbers,
surds and Pythagoras' theorem. I have been delighted when these
ideas have come out of the problem but basically my interest is in
the process of seeing the mathematics in the situation and posing
some problems, so the rest is just a bonus. Letting go is the key
requirement but the invitation to play and write ideas offers an
excellent opportunity for assessment too!

In Closing

In the end a problem is as good as we make it. There is a
requirement for patience, flexibility and allowing time to think
and share. But that is what mathematicians do. In fact that is what
we all do when we are stuck. We take a break, go and make a cup of
tea and even talk to a friend. We make sense of situations at
levels we can understand and then learn a little more by doing. I
think this can be summed up by saying:

Give learners space

Value their differences

Learn from what they do and help them to make connections

Use the inherent richness of opportunities to highlight
interesting mathematics

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.